13 MEASURES WITH BASE (JL 183 PROBLEMS 1. Show that the locally convex metrizable space L}Wt1n(X, p) (resp. L,1^, C(X, ^)) is complete (in other words, is a Frechet space). 2, Let 3? be a vector subspace of -^J^R containing the subspace Jf of ^-negligible functions. Suppose that the quotient space H = &l Jf is endowed with the structure of a real Hilbert space, in which the scalar product is denoted by (/ \§) and the norm J/j. Suppose moreover that, for every compact subset K of X, there exists a constant aK ^ 0 such that (A) for all ue Jf? (cf. Section 15.11, Problem 26). (We shall also use the notation (u\v) and|w|for (u\v) and|#|when u and v belong to .#".) Let ^g denote the set of bounded measurable functions on X with compact support. (a) Given any function /e &g, show that there exists a function Uf e Jf such that OLK| #) = [ufd\L for all w e #e\ and that the class of U7 in H is uniquely determined by the class of/. The function Uf is called the potential of/. Show that the set of classes of potentials Uf is dense in the Hilbert space H (use (6.3.2)). If f,g are two elements of J^g, then (U'1 IP) - (g\Jf dp = f /U' <//*. (b) As/runs through the functions ^0 belonging to ^fg, the set of potentials U7 is a convex cone in .#*. We denote by & the closure of this cone (with respect to the topology defined by the seminorm \u \ on .?f). The elements of ^ are called pure potentials. Let P denote the image of & in H. For each element u e H let 0 be the projection of u on P (Section 12.15, Problem 3(a)). Show that we have |#|2 — (0|#), that v(x) ^ u(x) almost everywhere, and that v is the only element of P satisfying these conditions. Also we have \v\ ^ \u\. Deduce that, for each # € H, the projection of 0 on the closed convex set of points v e H such that v(x) ^ u(x) almost everywhere belongs to P. (c) Deduce from (b) that an element v e 3f? belongs to & if and only if (v\w) i>0 for all w e 3? such that w(x) g; 0 almost everywhere (consider the difference between 0 and its projection on 0>). Equivalently, \v + vP| ^ \v\ for all w e jf such that w(x) > 0 almost everywhere. (d) Suppose that for all u e 3tf we have \u\ e tf and |(|w|)~ | ^ \tk\. Show that every pure potential u is ^0 almost everywhere (use (c)). If w, v are two pure potentials, show that inf(w, v) is a pure potential. (Among the elements of tf which majorize inf(w, «), consider an element w such that \w\ is a minimum; then w is a pure potential, by (b), and we have (u+w\u—w)<s(u+w\\u~ w>|). By calculating |(inf(w, w))~ |2, deduce that |(inf(«, w))~| <|#|; likewise that. |(inf(y, w))~\ ^ \w\; and hence that w — inf(«, w) = inf(t?, w) almost everywhere.) (e) With the same hypotheses as in (d), show that if/e JSfg5 is ;>0 almost everywhere, and if u E & is such that Uf(x) <; u(x) almost everywhere in the set of points xeX such that f(x) > 0, then Uf(x) < u(x) almost everywhere in X (*' principle of domina- tion")- (Observe that v = inf(Uf, u) is a pure potential, that (Uf | U/ — w) *• 0 and (v\ Uf — v)^> 0, and deduce that v = Uf almost everywhere.) (f) Suppose that the hypotheses in (d) are satisfied and also that for all u e # we have inf(w, 1) 6 3P and |(inf(w, 1))~ | g \u\. Show then that if w is a pure potential, and